# Subsets of vector spaces

i'm not sure if i'm posting this in the right place, so forgive me if i'm wrong! in my linear algebra revision i found that i'm struggling with one of the questions:
Let S and T be subsets of a vector space, V. Which of the following statements are true? Give a proof or a counterexample.
a) Sp(S n T)=Sp(S) n Sp(T)
b) Sp(S u T)=Sp(S) u Sp(T)
c) Sp(S u T)=Sp(S) + Sp(T)

For a) i've said its false, and my counter-example is S={(1,0)} and T={(0,1)} for R^2. But then I get LHS= Span of the empty set? Is this correct?
and from the answers i know that b)is false and c)is true but i have no idea how to prove this.

also just in general, what is the difference between "+" and "u" in this situation?

Any help would be appreciated! thank you!

## Answers and Replies

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Dick
Homework Helper
'+' here means vector sum. x is in A+B if x=a+b, where a is in A and b is in B. Your first counterexample is fine. 'u' is set union. Try applying the same S and T as in part a) in part b).

well if i tried it for b)

LHS = sp((1,0)u(0,1)) = the x axis or the y axis..?
RHS= sp(1,0) u sp(0,1) = the x axis or the y axis
so they're the same?

Dick
Homework Helper
well if i tried it for b)

LHS = sp((1,0)u(0,1)) = the x axis or the y axis..?
RHS= sp(1,0) u sp(0,1) = the x axis or the y axis
so they're the same?
Wrong on the LHS. (1,0)u(0,1) doesn't make any sense. You union sets, not vectors. Write that as SuT={(1,0)}u{(0,1)}={(1,0),(0,1)}, i.e. a set with two elements. What's the definition of 'sp'? Look it up if you have to.

HallsofIvy
By the way, $Sp(0, 1)\cup Sp(1, 0)$ is NOT empty- it is the 0 vector. The intersection of two vector spaces is always a vector space.